Found problems: 563
2002 AMC 12/AHSME, 23
The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$.
$\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$
2005 India Regional Mathematical Olympiad, 1
Let ABCD be a convex quadrilateral; P,Q, R,S are the midpoints of AB, BC, CD, DA respectively such that triangles AQR, CSP are equilateral. Prove that ABCD is a rhombus. Find its angles.
1982 Spain Mathematical Olympiad, 4
Determine a polynomial of non-negative real coefficients that satisfies the following two conditions:
$$p(0) = 0, p(|z|) \le x^4 + y^4,$$
being $|z|$ the module of the complex number $z = x + iy$ .
2000 Romania National Olympiad, 3
A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon.
Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.
1990 IMO Shortlist, 16
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
2000 Romania National Olympiad, 1
Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that:
$$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$
2017 District Olympiad, 4
Let $ C $ denote the complex unit circle centered at the origin.
[b]a)[/b] Prove that $ \left( |z+1|-\sqrt 2 \right)\cdot \left( |z-1|-\sqrt 2 \right)\le 0,\quad\forall z\in C. $
[b]b)[/b] Prove that for any twelve numbers from $ C, $ namely $ z_1,\ldots ,z_{12} , $ there exist another twelve numbers $ \varepsilon_1,\ldots ,\varepsilon_{12}\in\{-1,1\} $ such that
$$ \sum_{k=1}^{12} \left| z_k+\varepsilon_k \right| <17. $$
2022 AIME Problems, 4
Let $w = \frac{\sqrt{3}+i}{2}$ and $z=\frac{-1+i\sqrt{3}}{2}$, where $i=\sqrt{-1}$. Find the number of ordered pairs $(r, s)$ of positive integers not exceeding $100$ that satisfy the equation $i\cdot w^r=z^s$.
2011 District Olympiad, 2
[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle.
[b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication:
$$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$
1990 IMO, 3
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
2009 ISI B.Stat Entrance Exam, 7
Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $\frac{x}{2\sin 36^\circ}$.
2019 Jozsef Wildt International Math Competition, W. 44
We consider a natural number $n$, $n \geq 2$ and the matrices
\begin{tabular}{cc}
$A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}$
\end{tabular}
Show that$$\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)$$ $$=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)$$where $\epsilon \in \mathbb{C}\backslash \mathbb{R}$, $\epsilon^{n+1}=1$
1948 Putnam, A5
If $\xi_1,\ldots,\xi_n$ denote the $n$-th roots of unity, evaluate
$$\prod_{1\leq i<j \leq n} (\xi_{i}-\xi_j )^2 .$$
2001 India IMO Training Camp, 1
Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.
2008 Brazil Team Selection Test, 2
Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$
for any complex number $x.$
2016 Nigerian Senior MO Round 2, Problem 1
Let $a, b, c, x, y$ and $z$ be complex numbers such that $a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2}$. If $xy+yz+zx=1000$ and $x+y+z=2016$, find the value of $xyz$.
2018 AIME Problems, 6
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.
2011 ELMO Shortlist, 3
Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$.
[i]Evan O'Dorney.[/i]
2007 Serbia National Math Olympiad, 2
In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.
2020 AMC 12/AHSME, 23
How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that
$$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$$
then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$
2016 South East Mathematical Olympiad, 5
Let a constant $\alpha$ as $0<\alpha<1$, prove that:
$(1)$ There exist a constant $C(\alpha)$ which is only depend on $\alpha$ such that for every $x\ge 0$, $\ln(1+x)\le C(\alpha)x^\alpha$.
$(2)$ For every two complex numbers $z_1,z_2$, $|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)$.
2019 District Olympiad, 3
Let $a,b,c$ be distinct complex numbers with $|a|=|b|=|c|=1.$ If $|a+b-c|^2+|b+c-a|^2+|c+a-b|^2=12,$ prove that the points of affixes $a,b,c$ are the vertices of an equilateral triangle.
1989 IMO Shortlist, 10
Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that
\[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C}
\]
2015 China National Olympiad, 1
Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that
\[ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).\]
2021 JHMT HS, 8
For complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy $x^3 + cx + p(1 + qi) = 0$. Suppose $c$, $p$, and $q$ were chosen so that all three complex roots $x$ satisfy $\tfrac{5}{6} \leq \tfrac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \tfrac{6}{5}$, where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real part of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.